Kinetic isotope effects significantly influence intracellular
metabolite [superscript 13]C labeling patterns and flux
determination
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Citation
Wasylenko, Thomas M., and Gregory Stephanopoulos. “Kinetic
Isotope Effects Significantly Influence Intracellular Metabolite
[superscript 13]C Labeling Patterns and Flux Determination.”
Biotechnology Journal 8, no. 9 (August 5, 2013): 1080–1089.
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http://dx.doi.org/10.1002/biot.201200276
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Wiley Blackwell
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Author Manuscript
Biotechnol J. Author manuscript; available in PMC 2014 June 04.
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Published in final edited form as:
Biotechnol J. 2013 September ; 8(9): 1080–1089. doi:10.1002/biot.201200276.
Kinetic isotope effects significantly influence intracellular
metabolite 13C labeling patterns and flux determination
Thomas M. Wasylenko and Gregory Stephanopoulos
Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA,
USA
Abstract
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Rigorous mathematical modeling of carbon-labeling experiments allows estimation of fluxes
through the pathways of central carbon metabolism, yielding powerful information for basic
scientific studies as well as for a wide range of applications. However, the mathematical models
that have been developed for flux determination from 13C labeling data have commonly neglected
the influence of kinetic isotope effects on the distribution of 13C label in intracellular metabolites,
as these effects have often been assumed to be inconsequential. We have used measurements of
the 13C isotope effects on the pyruvate dehydrogenase enzyme from the literature to model
isotopic fractionation at the pyruvate node and quantify the modeling errors expected to result
from the assumption that isotope effects are negligible. We show that under some conditions
kinetic isotope effects have a significant impact on the 13C labeling patterns of intracellular
metabolites, and the errors associated with neglecting isotope effects in 13C-metabolic flux
analysis models can be comparable in size to measurement errors associated with GC–MS. Thus,
kinetic isotope effects must be considered in any rigorous assessment of errors in 13C labeling
data, goodness-of-fit between model and data, confidence intervals of estimated metabolic fluxes,
and statistical significance of differences between estimated metabolic flux distributions.
Keywords
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Isotope Effects; Isotopomer Modeling; Metabolic Engineering; Metabolic Flux Analysis;
Modeling errors
1 Introduction
Cells assimilate carbon, energy, and reducing power for growth and other essential
processes through the reactions of central carbon metabolism. The magnitudes of the fluxes
through these central metabolic reactions are tightly controlled, and represent the functional
output of transcriptional, translational, and post-translational regulatory processes [1, 2].
Consequently, in microbial systems knowledge of the distribution of metabolic fluxes in
© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
Correspondence: Prof. Gregory Stephanopoulos, Department of Chemical Engineering, Massachusetts Institute of Technology, Room
56-469C, 77 Massachusetts Ave, Cambridge, MA 02139 gregstep@mit.edu.
The authors declare no conflict of interest.
Wasylenko and Stephanopoulos
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different strains or under different conditions has proven to be extremely powerful in basic
physiological studies [3, 4].
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Moreover, experimentally determined metabolic flux distributions can be informative in
biotechnological applications, for instance in understanding the genetic manipulations and
metabolic rearrangements that might lead to increased production of a desired compound [5]
or how metabolism changes over the course of a fermentation [6]. In recent years, methods
for metabolic flux estimation have begun to be applied in more complex plant systems [7,
8], and in mammalian systems elucidation of metabolic fluxes has provided insight into the
proliferating phenotype of cancer cells [9] and identified a novel target for antiviral therapies
[10].
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Determination of intracellular metabolic fluxes is experimentally challenging. Direct
measurement of these metabolic fluxes is impossible so that mathematical models are
required to estimate the intracellular flux distribution from other directly measurable
quantities [11]. In the metabolite balancing approach, measured rates of consumption and
production of extracellular metabolites (referred to as “extracellular flux measurements”),
known stoichiometry of intracellular reactions, and a pseudo- steady state assumption on
intracellular metabolite concentrations are used to construct a mass balance constraint for
each metabolite in the metabolic network [12]. However, in most systems of interest the
number of unknown fluxes to be estimated exceeds the number of metabolites for which a
mass balance equation can be written so that the resulting system of linear equations is
underdetermined and an infinite number of solutions for the metabolic flux distribution
exist. Additional constraints can be obtained through the application of isotopic tracers,
substrates which are selectively “labeled” with rare, stable isotopes. The most commonly
used stable isotope label is 13C, and so we will focus on 13C-labeling experiments in this
contribution.
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The 13C atoms from the tracer may be incorporated into any of the carbon atom positions of
an intracellular metabolite. For a metabolite with n carbons in its backbone, there are 2n
possible carbon labeling patterns, each of which corresponds to a different combination
of 12C and 13C atoms at the various carbon atom positions. These 2n species are referred to
as the “isotopic isomers,” or “isotopomers,” of the metabolite. For each metabolite, the set
of 2n mole fractions describing the fraction of the total metabolite pool with each
possible 13C labeling pattern is referred to as the “isotopomer distribution” of that
metabolite. If an appropriate choice is made for the tracer, these isotopomer distributions
will be functions of the intracellular metabolic fluxes, so that information about metabolite
labeling patterns can be used to add constraints to the stoichiometric models described
above. Such information can in principle be obtained using 1H nuclear magnetic resonance
(NMR) [13], 13C NMR [11, 14, 15], or mass spectroscopy (MS) [16, 17], although in
practice MS has become the technology of choice for determination of 13C labeling patterns
because its sensitivity and precision greatly exceed those of NMR [18, 19]. The
supplementation of a stoichiometric model with additional constraints derived from
measured isotopic labeling patterns typically yields a system that is significantly
overdetermined, allowing for determination of a unique flux solution.
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Over the past 20 years, rigorous quantitative methods have been developed for the
estimation of the metabolic fluxes through central carbon metabolism using such
stoichiometric models augmented with 13C labeling data [11, 13, 20–24]. Flux estimation is
achieved through an iterative procedure in which a putative metabolic flux distribution is
generated and the 13C label distributions in intracellular metabolites that would result from
this flux distribution are predicted. The putative metabolic fluxes are refined until the
predicted 13C metabolite labeling data match as closely as possible the 13C labeling data
obtained from experiment. The metabolic flux distribution that minimizes the lack-of-fit
between simulated and measured 13C metabolite labeling data is considered to be the best
estimate for the true intracellular metabolic fluxes.
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The key step in these flux estimation algorithms is the prediction of the 13C labeling state
that will result from a putative flux distribution. This typically involves the solution of a
large set of isotopomer balance equations, or a similar set of equations with equivalent
information content [22, 25]. However, these isotopomer balance equations implicitly
assume the absence of isotope effects on the rates of the enzyme-catalyzed reactions of
central carbon metabolism. That is, these equations inherently assume that the enzymes
involved in central carbon metabolism will turn over all isotopomers of their substrate
metabolites at the same rate. There is in fact a large body of literature showing that isotope
effects do occur in central carbon metabolism. For instance it is well-known that plants
assimilating inorganic carbon preferentially assimilate 12C over 13C, and that the extent of
the discrimination against 13C depends on whether carbon is assimilated through Rubisco or
PEP carboxylase [26, 27]. More generally, 13C atoms form stronger bonds than 12C atoms
so that the presence of 13C atoms in a metabolite is expected to slow the rate of its
enzymatic conversion [16, 28]. Nonetheless in 13C-metabolic flux analysis (MFA) it has
commonly been assumed, either explicitly [16, 20, 29] or implicitly, that such carbon
isotope effects are negligible in the prediction of the 13C labeling states of intracellular
metabolites. To our knowledge, the validity of this assumption has never been investigated
quantitatively. Although Christensen and Nielsen [16] concluded that isotope effects were
unlikely to significantly affect 13C labeling patterns of intracellular metabolites, using gas
chromatography–combustion-isotope ratio mass spec-trometry (GC–C-IRMS) Heinzle et al.
have shown that isotope effects do significantly influence 13C labeling data in
specialized 13C-MFA experiments at low 13C enrichments, and that correction for these
isotope effects is necessary for flux estimation in these systems [28, 30]. In this contribution,
we sought to quantify the error that could result from neglecting carbon isotope effects in
traditional 13CMFA experiments, with standard levels of 13C enrichment and conventional
MS measurements used to obtain 13C labeling data for flux estimation. As a case study, we
investigated the potential for kinetic isotope effects to cause isotopic fractionation at the
pyruvate node.
2 Modeling isotopic fractionation at the pyruvate node
It has been known for more than 50 years that lipids and the carboxyl group of leucine, both
of which are derived from acetyl-CoA, are generally depleted in the 13C isotope relative to
other biomass constituents [31, 32]. Monson and Hayes [33] used indirect evidence from
measurement of positional 13C enrichment in lipids to conclude that this depletion of 13C
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was a result of an isotope effect on the reaction catalyzed by pyruvate dehydrogenase
(PDH), and the isotope effects associated with PDH enzymes from Escherichia coli and
Saccharomyces cerevisiae have been characterized by Melzer and Schmidt [34]. How ever,
the influence that these isotope effects will have on metabolite isotopomer distributions is
difficult to predict, as isotopic fractionation due to isotope effects will depend on the
metabolic flux distribution and the kinetic state of the system [35, 36]. We performed
simulations to investigate the potential consequences of isotope effects at the pyruvate node
in 13C-MFA labeling experiments.
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To model the influence of isotope effects at the pyruvate branch point on intracellular
metabolite isotopomer distributions, we constructed a simplified model of the pyruvate node
(Fig. 1). Pyruvate is produced from phosphoenolpyruvate in the reaction catalyzed by
pyruvate kinase (PK). We assume that this reaction is effectively irreversible, and the flux
through PK is denoted vPK. We further assume that all reactions other than the PDH reaction
that consume pyruvate can be lumped into a single reaction forming a metabolite X, as in
[33]. These reactions that form metabolite X could include anaplerotic reactions, amino acid
biosynthesis (e.g. synthesis of ala-nine), or production of byproducts such as lactate. We
further assumed that the reaction converting pyruvate to X is irreversible and has no
significant isotope effects. The latter assumption is justified by the fact that reactions in
which carbon–carbon bonds are broken (such as the PDH reaction) are generally expected to
have much larger 13C isotope effects than reactions in which all carbon–carbon bonds are
left intact (such as production of lactate or ala-nine from pyruvate). The flux through the
reaction converting pyruvate to metabolite X is denoted vX. Finally, for simplicity we
assumed that acetyl-CoA is synthesized exclusively from pyruvate through PDH. This
reaction is assumed to be irreversible, and the flux from pyruvate to acetyl-CoA is denoted
vPDH.
Here we define some nomenclature that will be used in the following discussion. The
positional isotopomers of pyruvate will be denoted ijk, where the binary variables i, j, and k
represent the labeling states at C1, C2, and C3 of pyruvate, respectively. These variables take
on values i, j, k = 0 if the carbon at the respective position in the pyruvate molecule is 12C
and i, j, k = 1 if the carbon is 13C. So for instance the isotopomer 001 has 12C atoms at C1
and C2 and a 13C atom at C3. The 13C labeling state of pyruvate can be described by a set of
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, where
is equal to the mole fraction of the total
eight isotopomer mole fractions
pyruvate pool with the labeling pattern ijk. The labeling state of the two-carbon unit of
acetyl-CoA will similarly be represented by four isotopomer mole fractions
.
At steady state, a mass balance on pyruvate dictates that the rate of pyruvate production
must equal the rate of pyruvate consumption:
(1)
Similarly, a mass balance can be written for each of the isotopomers of pyruvate:
(2)
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where
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is the rate at which pyruvate isotopomer ijk is produced through PK and
and
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are the rates at which pyruvate isotopomer ijk is metabolized to metabolite X and
acetyl-CoA, respectively. We define f to be the ratio of the flux through PDH to the flux
through PK (Fig. 1):
(3)
and the variables fijk are defined analogously for each of the isotopomers of pyruvate:
(4)
The total flux of pyruvate through PDH must equal the sum of the fluxes of each of the
pyruvate isotopomers:
(5)
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Using the definition of fijk, this can be rewritten as:
(6)
which is equivalent to:
(7)
where we have introduced the “isotopomer flux fraction”
:
(8)
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The isotopomer flux fraction
is equal to the fraction of the total flux through PK that
produces the pyruvate isotopomer ijk. We note that although under the typical assumption
that isotope effects are negligible the isotopomer flux fractions are equal to the pyruvate
isotopomer mole fractions
, in the case where isotope effects are present this is not the
case; the
will deviate from the
and Eq. (7) must be written as a flux fractionweighted average rather than as a mole fraction-weighted average.
In order to quantify the effects of isotopic fractionation at the pyruvate branch point, we
sought to determine the fijk. In the idealized case (without isotope effects),
(9)
i.e. the fijk are independent of i, j, and k. However, when isotope effects are present this will
not be the case.
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The split ratio at any branch point can be viewed as the ratio of two effective rate constants.
In the model system described above, there are two competing reactions which consume
pyruvate:
We assume the fluxes of these reactions can be approximated by Eqs. (10) and (11),
respectively:
(10)
(11)
where kPDH is the effective rate constant for the PDH reaction, kX is the effective rate
constant for the reaction forming metabolite X, and [Pyr] is the intracellular concentration of
pyruvate. Substituting Eq. (1) into Eq. (3), f can be expressed as:
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(12)
Substituting the expressions from Eqs. (10) and (11) into (12), we have:
(13)
so that f is expressed as a ratio of rate constants. The fijk can similarly be expressed as ratios
of rate constants:
(14)
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where
is the rate constant for the PDH reaction with the pyruvate isotopomer ijk as
the substrate. kX is constant for all isotopomers of pyruvate since we have assumed the
can be expressed relative to
isotope effects on this reaction are negligible. The
(which can be fixed to equal one). We define a set of βijk, which are equal to the ratios
so that the rate constants can be expressed as:
(15)
The βijk for singly labeled isotopomers of pyruvate can be determined directly from the data
of Melzer and Schmidt, who have measured the isotope effects on C1, C2, and C3 of
pyruvate. These are denoted α1, α2, and α3, respectively:
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(16)
Melzer and Schmidt [34] have determined these parameters to be α1 = 1.0093 for E. coli and
1.0238 for S. cere -visiae, α2 = 1.0213 for E. coli and 1.0254 for S. cerevisiae, and α3 =
1.0031 for the enzymes of both organisms.
The rates of reaction for isotopomers with multiple 13C atoms were not measured by Melzer
and Schmidt. Thus the rate constants for these isotopomers must be approximated from the
rate constants for the singly labeled isotopomers. We investigated two different cases in
which the βijk are defined by the following two equations:
(17)
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(18)
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Equation (17) assumes that the isotope effects are roughly additive. Each βijk is equal to the
product of the isotope effect contributions of the three individual carbon atoms of pyruvate,
where the isotope effect contribution of each individual carbon atom is equal to one (if the
carbon is 12C) or the reciprocal (multiplicative inverse) of its respective α-value (if the
carbon is 13C). The result is that multiply labeled isotopomers will have rate constants that
reflect the product of the isotope effects on each of their labeled atoms. Equation (18)
assumes that the isotope effects are not additive and that the 13C atom with the largest
isotope effect will dominate the kinetics. In this case, each βijk is equal to the minimum of
the individual carbon atom contributions, where the individual carbon atom contributions
are the same as above. Thus multiply labeled isotopomers will not have rate constants that
are smaller than those of the singly labeled isotopomers. This latter case will represent a
lower bound on the error that can be expected to result from neglecting isotope effects in the
enzymatic reactions of central carbon metabolism.
Under the assumptions listed above, the fractionation of isotopes at the pyruvate node (and
therefore the error associated with neglecting isotope effects) will depend solely on f and the
isotopomer flux fractions
. To determine realistic values for the
, we performed
simulations based on a flux distribution similar to the one estimated in [6] using five
commonly used glucose tracer mixtures and glucose labeled to natural abundance as the
substrates. For simplicity, we assumed that isotope effects in the metabolism of glucose to
are equal to the
obtained from
pyruvate were negligible. Under this assumption, the
simulation of the normal isotopomer balance equations. The selected tracers were 1-13Cglucose (commonly used to estimate flux through the oxidative pentose phosphate pathway
[37–39]), 1,2-13C2-glucose (determined to be the optimal tracer for a mammalian cell
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network in [40]), 3,4-13C2-glucose (determined to be optimal for resolution of pyruvate
carboxylase flux in [41]), 20% U-13C -glucose [37–39], and 75% 1-13C-glucose+25%
U-13C6-glucose (the tracer mixture used in the original study [6]). The isotopomer flux
fractions resulting from application of these tracers and natural abundance glucose given the
flux distribution in [6] and neglecting isotope effects in the metabolism of glucose to
pyruvate are summarized in Table1.
For a given value of f, the isotopomer distribution of the two-carbon unit in acetyl-CoA,
,
can be predicted for a specified tracer and the associated isotopomer flux fractions
in
the following way: the choice of the organism determines the values of α1, α2, and α3. The
βijk are then computed using either Eq. (17) or (18), depending on whether isotope effects
are assumed to be additive or not. The rate constants
are computed using Eq. (15)
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(with
fixed equal to one – the results are independent of the value of
) and
substituted into Eq. (14). Equations (14) and (7) then yield nine equations with nine
unknowns (kX and fijk), which can be solved using a nonlinear equation solver (we used the
Matlab function fsolve.m). Assuming there are no isotope effects on the reactions
downstream of acetyl-CoA that significantly affect its labeling pattern, the isotopomer
distribution of acetyl-CoA will then be given by:
(19)
In the idealized case (neglecting isotope effects), using Eq. (9) this can be simplified to:
(20)
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Predicted isotopomer flux fractions As mentioned above, 13C labeling data are often
obtained from MS, which resolves isotopomers by molecular mass. Consequently,
isotopomers with the same number of 13C atoms cannot be differentiated and are lumped
into a single “mass isotopomer,” which is simply a set of all the isotopomers of a metabolite
with a particular mass (or equivalently, a particular number of 13C atoms). For a metabolite
with n carbons and a mass M when all of the carbon atoms are 12C, MS will yield a set of n
+ 1 mass isotopomer mole fractions, where the ith mole fraction is equal to the mole fraction
of the total metabolite pool with mass M + i (i = 0, 1, 2, n). This set of mole fractions is
referred to as the “mass isotopomer distribution” (MID) of the metabolite. The mole fraction
of each mass isotopomer is equal to sum of the mole fractions of its constituent isotopomers;
for instance the M + 2 mole fraction of the pyruvate MID will be equal to the sum
. The set of MIDs obtained from MS (for as many metabolites as can be
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detected and accurately quantitated) makes up the 13C labeling dataset used in the flux
estimation algorithms described above. In order to estimate the error in MS 13C labeling
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data that could result from neglecting isotope effects, we converted both the
and
isotopomer distributions into MIDs:
(21)
where yAi is equal to the mole fraction of the M + i mass isotopomer in the MID of the
acetyl-CoA two-carbon unit. These equations hold for both the idealized case and the case
with isotope effects included. The errors associated with the assumption that isotope effects
(the MID mole fractions
are negligible can be estimated to be the difference between
expected to be obtained from a simulation which neglects isotope effects) and
MID mole fractions expected to be obtained from a 13C labeling experiment):
(the
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(22)
3 Results
We computed the
as functions of f for each of the five glucose tracers listed above
and for glucose labeled to natural abundance using the αi values measured by Melzer and
Schmidt [34] for both the E. coli and S. cerevisiae PDH enzymes. Two typical results for the
E. coli PDH enzyme assuming additivity of isotopes effects are shown in Fig.2.
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It can be seen that the magnitude of the error is maximum when f is small and decreases
linearly as f increases, approaching zero as f approaches unity. This is a well-known result
[33, 36]. There is zero error associated with isotope effects when f = 1 because isotopic
fractionation can only occur at a branch point in the metabolic network [35]. When
metabolism is at steady state, if one enzyme discriminates against an isotopomer of its
substrate metabolite then there must be another enzyme in the cell that turns over a
disproportionately high fraction of that isotopomer due to conservation of mass. For
instance, in the model system investigated above PDH discriminates against the heavy
isotopomers of pyruvate and consequently the enzyme forming metabolite X will metabolize
a disproportionately high fraction of these heavy isotopomers. If the conversion of the PDH
reaction is 100%, the PDH enzyme must turn over all isotopomers of pyruvate completely
and so there can be no discrimination against any of the pyruvate isotopomers at steady
state.
Under the assumption of additive isotope effects, the error associated with the E. coli PDH
enzyme with glucose labeled to natural abundance as the substrate approach- es 0.025 mol%
as f approaches zero (Fig.2). An identical result (to the nearest thousandth of a mole percent)
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is obtained if isotope effects are assumed to be non-additive (data not shown); because
metabolism of glucose labeled to natural abundance produces very few multiply labeled
isotopomers, the predicted errors with this substrate are relatively insensitive to the
assumption on additivity of isotope effects. The predicted maximum error of 0.025 mol% is
in good agreement with the results of Monson and Hayes, who predicted acetyl-CoA would
be depleted in 13C relative to pyruvate by approximately 23 per mille for small f (see Fig. 4
of [33]). When the 13C enrichment in glucose is increased by introduction of 13C-labeled
tracers, the errors in MID measurements associated with neglecting isotope effects increase.
The results with 20% U-13C6-glucose as the tracer indicate that these errors can approach
0.5 mol% for the E. coli PDH enzyme for small f.
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In order to investigate the effect of the organism (E. coli vs. S. cerevisiae), the glucose
tracer, and the assumption on the additivity of isotope effects for multiply labeled
isotopomers of pyruvate (Eqs. 17 and 18), we visualized the error in each case with f fixed to
a value of 1% (Fig.3). This represents something of a “worst case scenario” since the errors
are maximum for small values of f. The errors associated with isotope effects are predicted
to be larger in S. cerevisiae than in E.coli, which is expected since the α1 and α2 values
measured by Melzer and Schmidt were greater for the S. cerevisiae PDH. The expected
errors with a 1-13C-glucose tracer are relatively small because metabolism of this tracer
yields pyruvate primarily labeled at C3, which does not participate in the bond that is broken
in the PDH reaction and consequently has a relatively small isotope effect. The expected
errors with a 3,4-13C2-glucose tracer are also relatively minor. At first this may seem
surprising since metabolism of this tracer yields pyruvate primarily labeled at C1, and there
is a significant isotope effect on this carbon. However, the set of isotopomer flux fractions
that result from application of this tracer is dominated by a single isotopomer (the
isotopomer with labeling pattern 100). In fact, in this case 85% of the flux to pyruvate
produces the isotopomers with the labeling patterns 100, 010, or 110, and the unlabeled
isotopomer 000 accounts for only approximately 13% of the total flux to pyruvate. The
small flux producing the unlabeled isotopomer, which is the isotopomer turned over at the
highest rate by PDH, mitigates the effects of isotopic discrimination so that the errors
associated with neglecting isotope effects are smaller than might be expected. The
applications of the tracers 1,2-13C2-glucose, 20% U-13C6-glucose, and 75% 1-13C-glucose +
25% U-13C6-glucose result in the highest expected errors. Metabolism of each of these
tracers results in an unlabeled (000) isotopomer flux fraction >50% and significant flux
fractions for isotopomers labeled at one or both of the carbons with large isotope effects.
Consequently, with these tracers neglecting isotope effects is expected to result in relatively
large errors.
It can be seen from Fig.3 that, under the assumption that isotope effects are additive, the
errors associated with isotope effects can exceed 0.5 mol% and can even approach 0.8 mol
%. Antoniewicz et al. have shown that MIDs of amino acids labeled to natural abundance
can be measured to an accuracy of 0.4 mol% and with a precision of 0.2 mol% using gas
chromatography (GC) coupled to MS [42]. Thus, in some cases the errors associated with
isotope effects may exceed the measurement errors associated with the GC/MS instrument
and could in fact be the dominant source of error in the modeling of GC/MS MID data for
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flux estimation. Even in the case where isotope effects are assumed to be completely nonadditive, the errors associated with isotopic discrimination can approach the 0.4 mol% upper
bound for error in GC/MS data which was given in [42].
4 Discussion
It was noted above that the errors in Fig. 3 represent something of a “worst-case scenario.”
Because the magnitudes of the errors associated with neglecting isotope effects decrease as f
increases, in systems in which the majority of pyruvate is metabolized through PDH the
actual errors will be significantly smaller. For these systems, the ratio of the actual error to
the error shown in Fig 3 will be approximately equal to 1–f, since the magnitude of the error
decreases linearly as f increases and is equal to zero when f equals unity. However in many
cases the errors associated with isotope effects will be significant. For instance, a recent 13CMFA study in the A549 cancer cell line revealed that in these cells the majority of pyruvate
was metabolized to lactate so that in this system f was only approximately 10% [40]. Thus
the errors associated with the acetyl-CoA two-carbon unit MID in 13C-MFA studies in
cancer cells may approach the errors shown in Fig. 3.
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Moreover, the dependence of reaction kinetics on 13C-labeling is not unique to the PDH
enzyme. In fact, for enzymes catalyzing carbon–carbon bond-breaking reactions, significant
isotope effects appear to be the norm rather than the exception. Gleixner and Schmidt [43]
observed an isotope effect of 1.6% on C3 of fructose-1,6-bisphosphate (FBP) for FBP
aldolase from rabbit muscle (although the isotope effect on C4 was negligible), and Hermes
et al. [44] determined the isotope effect on C4 of malate to be approximately 3% for malic
enzyme from chicken liver. Significant isotope effects have also been observed for the
enzymes of the oxidative pentose phosphate pathway. Hermes et al. [44] determined the
isotope effect on C1 of glucose-6-phosphate (G6P) to be 1.65% for the G6P dehydrogenase
from Leuconostoc mesenteroides, while the isotope effect on C1 of 6-phosphogluconate
(6PG) has been estimated to be anywhere from 0.59 to 2.25% for 6PG dehydrogenases from
various sources [45–47]. The results for 6PG dehydrogenase (and the results for the PDH
enzymes of E. coli and S. cerevisiae discussed above) reveal that the isotope effect on a
single chemical reaction can vary significantly across different enzymes and different
organisms. In fact, the isotope effect associated with a single enzyme can in some cases be
highly condition-dependent – Grissom and Cleland measured the isotope effect on C6 of
isocitrate for NADP-dependent isocitrate dehydrogenase over a range of pH values and
found that the isotope effect varied from 0.3% (pH 7.0) to 2.76% (pH 4.1) [48]. The
presence of significant isotope effects in the kinetics of such a large number of the enzymes
of central carbon metabolism and the variation in the magnitudes of these effects across
different organisms and under different conditions makes it difficult to predict the impact
that isotopic discrimination will have on the 13C labeling patterns of intracellular
metabolites. However, given the preceding analysis (in which we neglected isotope effects
in all reactions upstream and downstream of PDH for simplicity) and the ubiquity of
enzymes with kinetics subject to isotope effects in the various pathways of central
metabolism, it seems likely that isotope effects will exert significant influence on the
distribution of 13C label in metabolic systems.
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Rigorous interpretation of 13C-MFA results requires an accurate assessment of the
measurement errors in the 13C labeling data. Knowledge of the measurement errors is
necessary to conduct goodness-of-fit tests, which quantify how well a set of labeling data fit
a metabolic model, and to compute confidence intervals for the estimated fluxes, which is
necessary to determine if differences in estimated fluxes in different strains or under
different conditions are statistically significant [23, 29, 49]. Failure to account for any
significant source of error in a 13C-MFA can potentially lead to underestimation of the
errors associated with labeling data. This can result in a biased estimated metabolic flux
distribution, failure of a chi-square test for goodness-of-fit even with a complete model
metabolic network, and overly optimistic estimation of the confidence intervals for the
computed fluxes. To-date, the most common methods for determining the magnitude of the
error in MS data have been measurement of MIDs of metabolites labeled to natural
abundance (for which the MIDs are theoretically independent of the flux distribution and
can be computed from the well-known natural abundances of heavy isotopes) [17, 42, 50],
calculation of standard deviations from replicate samples (which is really a measure of
precision, not accuracy) [5, 50], and comparison of labeling data for metabolites which
should in principle have identical labeling patterns (i.e. comparison of redundant data) [24].
None of these methods are capable of quantifying the errors associated with neglecting
isotope effects in central carbon metabolism. These errors are structural in nature and can
only be quantified by modeling the impact of isotope effects on the distribution of 13C atoms
in intracellular metabolites, which at present seems to be a challenging task. We further note
that as long as the isotopomer balance equations used for prediction of the labeling state
associated with a flux distribution neglect isotope effects, the errors associated with isotope
effects place a bound on the accuracy that can be achieved in modeling 13C labeling data.
No matter how accurate the measurement technique, the errors will never be smaller than
those associated with the assumption that isotope effects are negligible.
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In this contribution, we have modeled isotopic fractionation at the pyruvate node to
quantitatively assess the potential modeling errors associated with kinetic isotope effects.
We have also proposed a framework that can be used to estimate modeling errors associated
with isotope effects at other branch points in metabolism. Our results show that under some
conditions the modeling errors associated with isotope effects are significant and will be
comparable in size to the measurements errors associated with GC–MS. The quantification
of these errors will facilitate accurate interpretation of the results of future 13C-MFA studies.
Acknowledgments
We acknowledge support from United States Department of Energy grant DE-SC0008744. T.M.W. is supported by
the MIT/NIGMS Biotechnology Training Program.
Abbreviations
CoA
coenzyme A
GC
gas chromatography
MFA
meta bolic flux analysis
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MID
mass isotopomer distribution
MS
mass spectrometry
PDH
pyruvate dehydrogenase
PK
pyruvate kinase
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Figure 1.
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The pyruvate node model system. We assume that pyruvate is produced through pyruvate
kinase (PK) and consumed through one of two reactions. In the pyruvate dehydrogenase
(PDH) reaction, the bond between C1 and C2 of pyruvate is broken (represented by the
dashed line), yielding one molecule of CO2 and one acetyl-CoA two-carbon unit. This
reaction is subject to isotope effects, as measured by Melzer and Schmidt (see text). In the
other reaction, pyruvate is converted to a metabolite X. In this reaction all carbon–carbon
bonds in pyruvate remain intact, so we assume the isotope effects on this reaction are small
relative to those on the PDH reaction. The ratio of the flux through PDH to the flux through
PK is equal to f, while the ratio of the flux to metabolite X to the flux through PK is equal to
1 – f. Circles represent the carbon atoms of each metabolite.
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Figure 2.
Typical results for errors associated with neglecting isotope effects as a function of f. The
predicted modeling errors associated with neglecting isotope effects on the E. coli PDH
enzyme (assuming additivity of isotope effects for multiply labeled isotopomers of pyruvate)
with (A) natural abundance glucose and (B) 20% U-13C6-glucose substrates are plotted as a
function of the ratio of flux through PDH to flux through PK, f. The errors for each of the
three mass isotopomer mole fractions of the acetyl-CoA two-carbon unit MID are shown.
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Figure 3.
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Expected errors associated with isotope effects at low values of f. The predicted modeling
errors associated with neglecting isotope effects at a fixed value of f = 0.01 for different
combinations of organism, tracer, and assumption on the additivity of isotope effects for
multiply labeled isotopomers of pyruvate. (A) Predicted errors for each glucose substrate are
plotted for the cases of E. coli assuming additive isotope effects, (B) S. cerevisiae assuming
additive isotope effects, (C) E. coli assuming non-additive isotope effects, and (D) S.
cerevisiae assuming non-additive isotope effects. NA, natural abundance glucose; 1 =
1-13C-glucose; 1,2 = 1,2-13C -glucose; 3,4 = 3,4-13C2-glucose; 20%U = 20% U-13C6glucose; 75%1 + 25%U = 75% 1-13C-glucose + 25% U-13C6-glucose.
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Table 1
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Predicted isotopomer flux fractions
Isotopomer flux fractions
Tracers
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NA
1
1,2
3,4
20%U
75%1 + 25%U
~PK
v 000
0.968
0.706
0.606
0.132
0.754
0.518
~PK
v 001
0.011
0.243
0.069
0.005
0.019
0.187
~PK
v 010
0.011
0.021
0.015
0.043
0.013
0.016
~PK
v 011
1.2E–04
0.006
0.228
0.001
0.018
0.027
~PK
v 100
0.011
0.017
0.017
0.755
0.029
0.031
~PK
v 101
1.2E–04
0.005
0.039
0.009
0.002
0.008
~PK
v 110
1.2E–04
0.002
0.012
0.052
0.008
0.011
~PK
v 111
1.3E–06
5.0E–04
0.013
0.002
0.157
0.202
The isotopomer flux fractions
are predicted for each of the five glucose tracer mixtures listed in the text and for glucose labeled to natural
abundance. NA, natural abundance glucose; 1 = 1-13C-glucose; 1,2 = 1,2-13C2-glucose; 3,4 = 3,4-13C2-glucose; 20%U = 20% U-13C6-glucose;
75%1 + 25%U = 75% 1-13C-glucose + 25% U-13C6-glucose.
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